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DOTTORATO DI RICERCA IN
Ingegneria Strutturale e Idraulica
Ciclo XXIV
Settore Concorsuale di afferenza: 08/B2 Settore Scientifico disciplinare: ICAR 08
TITOLO TESI
Static analysis of functionally graded cylindrical and conical
shells or panels using the generalized unconstrained third
order theory coupled with the stress recovery
Presentata da: Luigi Rossetti
Coordinatore
Dottorato
Relatore
Chiarissimo Prof. E. Viola
Chiarissimo Prof. E. Viola
Index
Chapter 1.………...p.1
Sommario ……….p.1 1.1 General literature trends……… .p.2 1.2 The aim of the present work………...p.3 1.3 Problem formulation ………..p.3 1.3.1 Third order displacement expansion………... p.4 1.3.2 Relations between strains and displacements……….p.5 1.3.3 Relations between stresses and strains……….p.7 1.3.4 Internal forces and moment resultants……….p.9 1.3.5 Normal and shear forces………p.10 1.3.6 Moments………...p.11 1.3.7 Higher order moments………...p.13 1.3.8 Shear forces………...p.14 1.3.9 Equilibrium equations………...p.15 1.3.9.1 The first fundamental equilibrium equation………...p.22 1.3.9.2 The second fundamental equilibrium equation………...p.24 1.3.9.3 The third fundamental equilibrium equation………..p.27 1.3.9.4 The fourth fundamental equilibrium equation………....p.30 1.3.9.5 The fifth fundamental equilibrium equation………...p.33 1.3.9.6 The sixth fundamental equilibrium equation………..p.35 1.3.9.7 The seventh fundamental equilibrium equation………..p.38 1.4 Equilibrium equations for doubly curved shells………...p.40 1.4.1 Stress recovery via GDQ………...p.43 Figures……….p.44 References………...p.45
Chapter 2.……….p.50
Sommario ………...p.50 2.1. Introduction………. p.51 2.2. Functionally graded composite cylindrical shell and fundamental system……….p.54 2.2.1 Fundamental hypotheses………p.54 2.2.2 Displacement field and constitutive equations………..p.55 2.2.3 Forces and moments resultants………..p.58 2.2.3.1 Normal and shear forces………p.59 2.2.3.2 Moments……….p.59 2.2.3.3 Higher order moments………p.60 2.2.3.4 Shear Forces………p.61 2.2.3.5 Higher order shear resultants………...p.61 2.2.4 Equilibrium equations………p.62 2.3 Discretized equations and stress recovery………p.65 2.4. Numerical results……….p.67 2.4.1 Classes of graded materials………p.67
2.4.2 Stress profiles of FGM1(1,0,0,p)cylindrical panels……….p.70
2.4.2.1 Generalized and traditional unconstrained theories………p.70
2.4.3 Stress profiles of FGM1(1,1,4,p)cylindrical shells……….p.71
2.4.3.1 Generalized unconstrained third and first order theories………p.71
2.4.4 Stress profiles of FGM1(1,0.5,2,p)cylindrical shells……….p.71
2.4.4.1 Generalized and traditional unconstrained theories………p.71 2.4.5 Stress profiles of FGM1(a ,0.2,3,2) and FGM2(a ,0.2,3,2) cylindrical panels………p.72 2.4.5.1 The generalized unconstrained theory………..p.72
2.4.6 Stress profiles of FGM1(1,0.5,c,2) cylindrical panels………...p.72
2.4.6.1 Generalized unconstrained first and third order theories………p.72 2.4.7 The stress recovery approach for the generalized unconstrained first and third order
theories………p.73 2.5 Literature numerical examples worked out for comparison………p.73 2.6 Final remarks and conclusion………...p.75 References………...p.77 Figures………p.83 Tables………..p.93
Appendix………..p.104
Chapter 3.………...p.109 Sommario ……….p.109 3.1 Introduction………p.110 3.2 Functionally graded composite conical shells and fundamental systems………...p.117 3.2.1 Fundamental hypotheses………..p.117 3.2.2 Displacement field and constitutive equations………p.118 3.2.3 Forces and moments resultants………p.121 3.2.3.1 Normal and shear forces ………..p.122 3.2.3.2 Higher order moments……….p.123 3.2.3.3 Shear forces ………p.124 3.2.3.4 Higher order shear resultants………...p.125 3.2.4 Equilibrium equations……….p.125 3.3 Discretized equations and stress recovery………..p.128 3.4 Stress profiles ……….p.132 3.4.1 The reference configuration……….p.134 3.4.1.1 The influence of the initial curvature effect with the semi vertex angle……….p.135 3.4.1.2 The influence of the initial curvature effect with the p - power exponent………p.136 3.4.1.2.1 Comparisons between the first and third order stress responses with the initial curvature effect and the p-power exponent………...p.136 3.4.1.3 The influence of the initial curvature effect with the a – material coefficient………….p.136 3.4.1.3.1 Comparisons between the first and third order stress responses with the initial curvature effect and the a-material coefficient……….p.137 3.4.1.4 Comparisons between the first and third order stress responses with the initial curvature effect and the b-material coefficient……….p.137
3.4.1.5 The influence of the L h/ aspect ratio with the - angle………p.138
3.4.1.5.1 The influence of the L h/ aspect ratio with the - angle………..p.138
3.4.1.6 Comparisons between the first and third order recovered and un-recovered transverse stress distributions………..p.138 3.4.1.7 The influence of boundary conditions ………...p.139
3.5 Comparison study ………..p.140 3.6 Conclusion………..p.140 References……….p.141 Figures………...p.148 Tables………p.167 Appendix………...p.171
Abstract
A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded conical and cylindrical shells subjected to mechanical loadings. Several types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the conical or cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally conical and cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.
Chapter 1
Third order Shear Deformation Theory
Sommario
Dopo aver analizzato lo stato dell’arte, si è fatta strada l’idea di sviluppare una teoria generale di deformazione a taglio del terzo ordine di tipo svincolato per gusci/pannelli di rivoluzione a doppia curvatura, costituiti da uno strato singolo di materiale a stratificazione graduale. Si è operata la scrittura del modello cinematico a sette parametri indipendenti, delle relazioni tra deformazioni e spostamenti arricchite dell'effetto della curvatura, delle equazioni costitutive per una lamina singola in materiale a stratificazione graduale e delle caratteristiche di sollecitazione in funzione degli spostamenti. Definiti i carichi esterni uniformi di natura trasversale, assiale e circonferenziale, è stato applicato il principio degli spostamenti virtuali per ricavare le equazioni indefinite di equilibrio e le condizioni al contorno. Pertanto si è proceduti alla scrittura della equazioni fondamentali con la sostituzione delle relazioni delle azioni interne espresse in funzione degli spostamenti, nelle equazioni indefinite di equilibrio. Compiuta la scrittura del sistema fondamentale si è pervenuti alla soluzione di esso in termini delle sette variabili di spostamento indipendenti, applicando la tecnica di quadratura differenziale di tipo generalizzato in tutti i punti della superficie di riferimento del panello/guscio. Dunque è stato possibile determinare le tensioni membranali in un punto arbitrario appartenente alla superficie di riferimento del panello/guscio ed elaborare poi la distribuzione di esse lungo lo spessore dell'elemento strutturale. Successivamente con il fine di pervenire alla determinazione completa del tensore delle tensioni, ovvero delle tensioni trasversali normale e tagliante, si è operata l'integrazione delle equazioni indefinite di equilibrio sfruttando la conoscenza delle tensioni membranali, determinate indirettamente dal sistema fondamentale, sempre utilizzando il metodo generalizzato di quadratura differenziale. Pertanto si è pervenuti alla determinazione dei profili di tensione trasversale normale e tagliante lungo lo spessore del panello/guscio. In ambito letterario, il percorso proposto ha degli attributi di autenticità in quanto consente di calcolare profili di tensione trasversale che soddisfano al pieno le condizioni al contorno, anche in presenza di carichi taglianti alle superfici di estremità. In tal modo viene superato uno dei limiti propri della teoria di Reddy che diversamente ritiene nulli a priori i carichi taglianti alle estremità del panello/guscio.
1.1 General literature trends
Two significant classes of two dimensional shell theories can be found in literature: the first based on the assumed form of the displacement field and the second based on the assumed form of the stress field. In both cases, the displacement or stress fields are expanded in increasing powers of the thickness coordinate. Nevertheless, displacement – based theories are more recurrent because they do not require the strain/stress compatibility condition in addition to the kinematic and equilibrium equations. It is proved that a third order expansion of the displacement field is optimal because it gives quadratic variation of transverse strains and stresses, and require no “shear correction factors” compared to the first order theory, where the transverse strains and stresses are constant through the shell thickness. A brief overview of research done in third order shell theories is also included in here.
The simplest and oldest plate theory is the classical Kirchhoff plate theory [1]. The so called Kirchhoff hypothesis includes the following assumptions: straight lines remain perpendicular to the reference surface and inextensible after deformation. In this manner both transverse shear and normal strains [2,3] are neglected. These assumptions in the model simplify the three dimensional problem to a two dimensional one and the governing equations are expressed in terms of three displacements of a point on the midsurface. Moreover the theory does not qualify to be called first order because the first order terms or rotations are not independent of the transverse displacement component. The theory is very useful in a wide range of problems when thickness is very small (two orders of magnitude less than the smallest in plane dimension). Transverse shear strains are also negligible.
The simplest first order shear deformation shell theory (FSDT) often referred to as the Mindlin plate theory [4-6], is based on the displacement expansion till to the first order, where the first order terms are the rotations of a transverse normal line and are independent of the transverse displacement component. The first idea of such expansion can be found in earlier works by Basset [7], Hencky [8] and Hildebrand et al. [9]. The normality is not invoked and in this way the rotation are independent of membrane and transverse displacement components and the transverse shear strains are non zero but independent of out of plane coordinate. This leads to the introduction of shear correction factors in the evaluation of the transverse shear forces.
Second order and higher order theories relax the Kirchhoff hypothesis further by allowing the straight lines normal to the midsurface before deformation to become curves. Second order shell theories are not so diffused because they also require shear correction factors.
Several third order plate theories have been developed by different researchers [10-24] but as pointed by Reddy [21] some of them are claimed to be new whereas they are not new, but only different in the form of the displacement expansions adopted.
Reddy [19,20] is the first one to develop the equilibrium equations of a third order shell theory with vanishing tractions for composite structures, using the principle of virtual displacements. By means of these assumptions, Reddy’s theory reduces the independent displacement components from seven to five. The theory leads to the accurate reconstruction of the effective transverse shear components but it excludes the presence of transverse shear loads on the boundary surfaces of the shell.
1.2 The aim of the present work
In the present work, by moving from Leung’s idea [25] a third order shear deformation theory has been developed by neglecting the Reddy’s assumptions. The present third order model involves seven unknown independent parameters and it includes the possible presence of shear uniform loads in addition to the normal uniform one on the extreme surfaces of composite shell. As in the Reddy’s theory no correction factor is introduced.
The third order shear deformation theory under discussion is formulated for a single lamina doubly curved shell of functionally graded material. The seven independent fundamental equations are achieved by applying the principle of virtual displacements and the fundamental system is solved by means of the GDQ method [26-62]. By using the GDQ solution in term of the generalized displacements of points on the reference surface, the membrane profiles of normal and shear stresses are determined throughout the thickness direction. Then, by considering the three dimensional equilibrium equations, by discretizing them via the GDQ method and by the knowledge of the membrane stress components, the transverse profiles of normal and shear stresses are determined with satisfaction of the boundary conditions at the extreme surfaces. The Reddy’s model lead to accurate transverse stress profiles by supposing the null values of transverse shear stress component at the extreme surfaces, whereas the present one in conjunction with the stress recovery from the three dimensional equations leads to accurate transverse shear stress profiles even if shear uniform loadings are present on the boundary surfaces.
1.3 Problem formulation
In this study, a single lamina doubly curved shell of functionally graded material represents the basic configuration of the problem (Fig.1). , s are the coordinates along the meridian and circumferential directions of the reference surface, respectively. The third orthogonal coordinate to
the middle plane along the shell normal is . - coordinate defines the distance of each point from the shell mid surface h 2 h 2 and h is the thickness of the shell. The angle between the extended normal n to the reference surface and the axis of rotation x3, or the geometric axis
3
xof the meridian curve, is defined as the meridian angle . The angle formed by the parallel circle
0( )
R and the x1 axis is designated as the circumferential angle . The meridian curves and the parallel circles are represented by the parametric coordinates (,s) upon the middle surface of the shell. The curvilinear abscissa s
of a generic parallel is related to the circumferential angle bythe relation sR0. The horizontal radius R0( ) of a generic parallel of the shell represents the distance of each point from the axis of revolution x3. Rb is the shift of the geometric axis of the curved meridian x3 with reference to the axis of revolution x3. The curvature radius Rfor a shell of revolution is defined by the relation R R0 sin. For a general shell of revolution, R R, ,
0
R are all independent of the -angle. The well known equation of Gauss - Codazzi is also considered : dR d0 R cos.
The position of an arbitrary point within the shell material is defined by the coordinates (0 ), 1 s(0 ) upon the middle surface, and s s0 directed along the outward normal and measured from the reference surface (h 2 h 2). In the present shell theory, the following assumptions are taken under consideration in the formulation: (1) the shell deflections are small and the strains are infinitesimal; (2) the transverse shear deformation is considered to influence the governing equations. In this manner the normal lines to the reference surface of the shell before deformation do not remain straight and normal after deformation; (3) the transverse normal strain is inextensible so that the normal strain is equal to zero; (4) the shell is moderately thick so that the transverse normal stress could be considered negligible; (5) the linear elastic behavior of composite materials is assumed; (5) the initial curvature effect is also taken into account.
1.3.1 Third order displacement expansion
Consistent with the assumptions of a moderately thick shell theory reported above, the displacement field considered in this study is that of the Third order Shear Deformation Theory and can be put in the following form :
where u, u ,s w are the displacement components of points lying on the reference surface ( ) 0 of the shell, along meridional, circumferential and normal directions, respectively. and s are normal to mid-surface rotations, respectively. and s are the higher order terms. The kinematic
hypothesis expressed by Eq.(1) is enriched by the statement that the shell deflections are small and strains are infinitesimal, that is w
,s h.1.3.2 Relations between strains and displacements
The relations of strains for a revolution shell are the followings [64]:
1 1 U W R R (2) 0 0 1 cos sin sin 1 U U W R R (3)
By considering , Eq.(3) can be written in the following form: s R0
0 0 1 cos sin 1 s s U U W s R R R (3.1) n W (4) 1 1 1 1 n U W R R R R R R (5) 0 0 0 1 1 1 1 n U W R R R R R R (6)
0 0 1 1 1 1 s sn U W R s R R R R (6.1) 0 1 1 cos 1 1 U U U R R R R (7)
By considering , Eq.(7) can be written in the following form: s R0
0 1 1 cos 1 1 s s s U U U s R R R R (7.1)
By substituting Eq.(1) in Eqs.(2-7.1), relations between strains and displacements become:
1 1
3 u w R R (8)
0 3cos sin cos
1 1 cos u u w R R (9)
By considering , Eq.(9) can be written in the following form: s R0
3 0 0 00
cos sin cos
1 1 cos s s s s u u w s R R s R R s R (9.1) 2 1 1 1 w
0 2 3 1 1 1 ( 3 2 ) 1 n w u R R R R (11)By considering , Eq.(11) can be written in the following form: s R0
1
( 1 3 2 2 3 ) 1 s sn s s s w u R s R R (11.1)
3 3 0 1 1 1cos cos cos
1 u R R u u R R (12)
By considering , Eq.(12) can be written in the following form: s R0
3 3 0 0 0 1 11 cos cos cos
1 s s s s s s s u R R u u s R s R s R R (12.1)
The transverse normal strain is n as in the assumptions. 0
1.3.3 Relations between stresses and strains
Relations between stresses and strains for a single lamina functionally graded shell are as follows:
11 12 12 22 66 44 55 0 s s s n s s n n sn sn Q Q Q Q Q Q Q (13) where [40,41]:
11 22 2 12 2 66 44 55 ( ) , 1 ( ) 1 2(1 ( )) E E Q Q Q E Q Q Q (14)The material properties of the functionally graded lamina vary continuously and smoothly in the thickness direction and are functions of volume fractions of constituent materials. Young’s modulus E( ) , Poisson’s ratio and mass density
of the functionally graded lamina
can be expressed as a linear combination of the volume fraction:
( ) ( ) ( ) C M C M C M C M C M C M V E E E V E V (15)where VC
is the volume fraction of the ceramic constituent material, while C,E ,C C andM
,E ,M M represent mass density, Young’s modulus, Poisson’s ratio of the ceramic and metal constituent materials, respectively.
In this work, the ceramic volume fraction VC
follows two simple four parameter power lawdistributions[40,41]: 1,2( , , , ) 1 1 : ( ) 1 2 2 p c a b c p C FGM V a b h h (16)
where the volume fraction index p ( 0 ) and the parameters p a, b, c determine the material variation profile along the thickness direction. The elastic engineering constants are written as follows:
2 2 3 4 5 6 7 8 9 , , , , , , , 2 , , (1, , , , , , , , , ) h ij ij ij ij ij ij ij ij ij ij ij h A B D E F L H M N V Q d
(17)1.3.4 Internal forces and moment resultants
Normal forces, moments, and higher order moments, as well as the shear force and higher order shear force are all defined by the following expressions:
2 3 2 , , (1, , ) 1 h h N M P d R
(18)
2 3 2 , , (1, , ) 1 h s s s s h N M P d R
(19)
2 3 2 , , (1, , ) 1 h s s s s h N M P d R
(20)
2 3 2 , , (1, , ) 1 h s s s s h N M P d R
(21)
2 2 3 2 , , (1, , ) 1 h n h T Q S d R
(22) 2 2 3 2 ( , , ) (1, , ) 1 h s s s sn h T Q S d R
(23)By considering the effect of the initial curvature in the formulation, the stress resultants
, ,
s s s
N M P are not equal to the stress resultants Ns,Ms,Ps, respectively. This assumption derives from the consideration that the ratios / R, / R are not neglected with respect to unity. The effect of initial curvature is characterized by the following coefficients as firstly done by Toorani Lakis [63]and then improved by Tornabene [55]:
1 2 3 2
0 0 0
2
1 2 3 2
0 0 0 0 0
sin 1 1 sin 1 1 sin 1
, ,
1 sin sin sin 1 sin sin 1
, , a a a R R R R R R R R b b b R R R R R R R R (24)
1.3.5 Normal and shear forces
By substituting Eqs.(13) in Eqs.(18-21), the following expressions are obtained:
11 1 11 2 11 3 11 12 12 0 12 11 1 11 2 11 3 11 0 11 1 11 2 11 3 11 12 0 12 11 1 11 2 11 3 11 12 0 12 1 cos sin 1 1 cos 1 cos s s s u u N A a B a D a E A u A R R s A w A a B a D a E w R R B a D a E a F B R R B E a F a L a H E s R R E s (25)
12 22 1 22 2 22 3 22 0 22 1 22 2 22 3 22 22 1 22 2 22 3 22 12 0 12 22 1 22 2 22 3 22 0 22 1 22 2 22 3 22 12 1 cos sin 1 1 cos 1 cos s s s u N A A b B b D b E u R R u A b B b D b E s A b B b D b E w A w R R B B b D b E b F R R B b D b E b F s E R R
22 1 22 2 22 3 22 0 22 1 22 2 22 3 22 s E b F b L b H E b F b L b H s (26)
66 66 1 66 2 66 3 66 66 0 66 66 1 66 2 66 3 66 66 0 66 66 1 66 2 66 3 66 66 0 1 cos 1 cos 1 cos s s s s s s s u u N A A a B a D a E A u s R R B B a D a E a F B s R R E E a F a L a H E s R R (27)
66 1 66 2 66 3 66 66 66 1 66 2 66 3 66 66 1 66 2 66 3 66 0 66 66 1 66 2 66 3 66 0 66 1 66 2 66 3 66 66 1 cos 1 cos 1 s s s s s s u u N A b B b D b E A s R A b B b D b E u B b D b E b F R s B B b D b E b F R R E b F b L b H E s R
66 1 66 2 66 3 66
0 cos s E b F b L b H R (28) 1.3.6 MomentsBy substituting Eqs.(13) in Eqs.(18-21), the following expressions are obtained:
11 1 11 2 11 3 11 12 12 0 12 11 1 11 2 11 3 11 0 11 1 11 2 11 3 11 12 0 12 11 1 11 2 11 3 11 12 0 12 1 cos sin 1 1 cos 1 cos s s s u u M B a D a E a F B u B R R s B w B a D a E a F w R R D a E a F a L D R R D F a L a H a M F s R R F s (29)
12 22 1 22 2 22 3 22 0 22 1 22 2 22 3 22 22 1 22 2 22 3 22 12 0 12 22 1 22 2 22 3 22 0 22 1 22 2 22 3 22 12 1 cos sin 1 1 cos 1 cos s s s u M B B b D b E b F u R R u B b D b E b F s B b D b E b F w B w R R D D b E b F b L R R D b E b F b L s F R R
22 1 22 2 22 3 22 0 22 1 22 2 22 3 22 s F b L b H b M F b L b H b M s (30)
66 66 1 66 2 66 3 66 66 0 66 66 1 66 2 66 3 66 66 0 66 66 1 66 2 66 3 66 66 0 1 cos 1 cos 1 cos s s s s s s s u u M B B a D a E a F B u s R R D D a E a F a L D s R R F F a L a H a M F s R R (31)
66 1 66 2 66 3 66 66 66 1 66 2 66 3 66 0 66 1 66 2 66 3 66 66 66 1 66 2 66 3 66 0 66 1 66 2 66 3 66 66 1 cos 1 cos 1 s s s s s u M B b D b E b F s u B B b D b E b F u R R D b E b F b L s D D b E b F b L R R F b L b H b M s F R
66 1 66 2 66 3 66
0 cos s s F b L b H b M R (32)1.3.7 Higher order moments
By substituting Eqs.(13) in Eqs.(18-21), the following expressions are obtained:
11 1 11 2 11 3 11 12 0 12 12 11 1 11 2 11 3 11 0 11 1 11 2 11 3 11 12 0 12 11 1 11 2 11 3 11 12 0 12 1 cos sin 1 1 cos 1 cos s s s u P E a F a L a H E u R R u E E w E a F a L a H w s R R F a L a H a M F R R F H a M a N a V H s R R H s (33)
12 22 1 22 2 22 3 22 0 22 1 22 2 22 3 22 22 1 22 2 22 3 22 12 0 12 22 1 22 2 22 3 22 0 22 1 22 2 22 3 22 12 1 cos sin 1 1 cos 1 cos s s s u P E E b F b L b H u R R u E b F b L b H s E b F b L b H w E w R R F F b L b H b M R R F b L b H b M s H R R
22 1 22 2 22 3 22 0 22 1 22 2 22 3 22 s H b M b N b V H b M b N b V s (34)
66 66 1 66 2 66 3 66 66 0 66 66 1 66 2 66 3 66 66 0 66 66 1 66 2 66 3 66 66 0 1 cos 1 cos 1 cos s s s s s s s u u P E E a F a L a H E u s R R F F a L a M a N F s R R H H a M a N a V H s R R (35)
66 1 66 2 66 3 66 66 66 1 66 2 66 3 66 0 66 1 66 2 66 3 66 66 66 1 66 2 66 3 66 0 66 1 66 2 66 3 66 66 1 cos 1 cos 1 s s s s s s u u P E b F b L b H E s R E b F b L b H u R F b L b H b M F s R F b L b H b M R H b M b N b V H s R
66 1 66 2 66 3 66
0 cos s H b M b N b V R (36) 1.3.8 Shear forcesBy substituting Eqs.(13) in Eqs.(22,23), the following expressions are obtained:
44 1 44 2 44 3 44 44 1 44 2 44 3 44 44 1 44 2 44 3 44 44 1 44 2 44 3 44 44 1 44 2 44 3 44 1 1 2 3 T A a B a D a E u R w A a B a D a E R A a B a D a E D a E a F a L E a F a L a H R (37)
55 1 55 2 55 3 55 0 55 1 55 2 55 3 55 55 1 55 2 55 3 55 55 1 55 2 55 3 55 55 1 55 2 55 3 55 0 sin 2 sin 3 s s s s s T A b B b D b E u R w A b B b D b E A b B b D b E s D b E b F b L E b F b L b H R (38)
44 1 44 2 44 3 44 44 1 44 2 44 3 44 44 1 44 2 44 3 44 44 1 44 2 44 3 44 44 1 44 2 44 3 44 1 1 2 3 w Q D a E a F a L u D a E a F a L R R D a E a F a L F a L a H a M L a H a M a N R (39)
55 1 55 2 55 3 55 55 1 55 2 55 3 55 0 55 1 55 2 55 3 55 55 1 55 2 55 3 55 55 1 55 2 55 3 55 0 sin 3 2sin s s s s s w Q D b E b F b L u D b E b F b L R s D b E b F b L F b L b H b M F b L b H b M R (40)
44 1 44 2 44 3 44 44 1 44 2 44 3 44 44 1 44 2 44 3 44 44 1 44 2 44 3 44 44 1 44 2 44 3 44 1 1 2 3 S E a F a L a H u R w E a F a L a H R E a F a L a H L a H a M a N H a M a N a V R (41)
55 1 55 2 55 3 55 55 1 55 2 55 3 55 0 55 1 55 2 55 3 55 55 1 55 2 55 3 55 55 1 55 2 55 3 55 0 sin 3 2sin s s s s s w S E b F b L b H u E b F b L b H R s E b F b L b H L b H b M b N H b M b N b V R (42) 1.3.9 Equilibrium equationsHere we use the principle of virtual displacements to derive the equilibrium equations consistent with the displacement field equations (1). The principle of virtual displacements can be stated in analytical form as:
2 2 ( ) 0 h s s s s n n sn sn s s h n s s s s d d p u R d ds p u R d ds p wR d ds m R d ds m R d ds r R d ds r R d ds
(43) where: 0 1 1 d R d R d R R (43.1)and p,p p m m r rs, n, , s, , s are the external uniform loadings applied on the reference surface.
By introducing Eqs.(8-12.1;13) into Eq.(43) and considering Eqs.(18-23), the following terms of the integral can be separated as follows:
2 2 0 0 0 0 h h d u N R d d N w R d d M R d d P R d d
(43.2)
2 2 cos sin cos cos h h u d N R d d N u R d d N w R d d M R d d M R d d P R d d P R d d
(43.3)
2 0 0 0 2 cos ( ) ( cos ) h h u d N R d d M R d d P R d d u N R d d N u R d d M R d d M R d d P
R d d P( cos ) R d d
(43.4)
2 0 0 0 2 0 0 ( ) 3 2 h n n h w d T u R d d T R d d T R R d d Q R R d d S R d d
(43.5)
2 0 2 0 sin 3 2 (sin ) h n n h w d T u R d d T R d d T R R d d Q R R d d S R d d
(43.6)By solving the integrals by parts in Eqs.(43.2-43.6), the resulting expressions are obtained:
0
0 0 N R u N R d d N R u u d d
(43.7)
0
0 0 M R M R d d M R d d
(43.8)
0
0 0 P R P R d d P R d d
(43.9)
N R
u N R d d N R u u d d
(43.10)
M R
M R d d M R d d
(43.11)
P R
P R d d P R d d
(43.12)
0
0 0 N R u N R d d N R u u d d
(43.13)
0
0 0 M R M R d d M R d d
(43.14)
0
0 0 P R P R d d P R d d
(43.15)
N R
u N R d d N R u u d d
(43.16)
M R
M R d d M R d d
(43.17)
P R
P R d d P R d d
(43.18)
0
0 0 T R w T R d d T R w wd d
(43.19)
T R
w T R d d T R w wd d
(43.20) By setting the coefficients of u, us, w, , s, , s to zero separately, the equilibrium equations are obtained:u : 0 1 cos 0 s s N N N N T p R s R R (44) s u : 0 0 1 cos sin 0 s s s s s s N N N N T p s R R R (45) w : 0 0 1 cos sin 0 s s n T T N T N p R s R R R (46) :
0 1 cos 0 s s M M M M T m R s R (47) s : 0 1 cos 0 s s s s s s M M M M T m R s R (48) : 0 1 cos 3 2 0 s s P P P P S Q r R s R R (49) s : 0 0 1 sin cos 3 2 0 s s s s s s s P P P P Q S r R s R R (50)It is worth noting that Eqs.(44-50) are derived by taking into account the definitions (18-23) of forces and moment resultants. The first three Eqs.(44,45,46) express the translational equilibrium along the meridional , circumferential s, and normal direction, respectively. The last four Eqs.(47,48,49,50) are rotational equilibrium equations about the s and directions, respectively. In particular, the first two are the effective rotational equilibrium equations, whereas the second two represent fictitious equations, which are derived by the computation of the additional terms of displacement.
Then, substituting the expressions (25-42) for the in-plane meridional, circumferential, and shearing force resultants N,N Ns, s,Ns, the analogous couples M,M Ms, s,Ms,P P P, s, s,Ps and the transverse shear force resultants T T Q Q S S, s, , s, , s, Eqs.(44-50) yield the fundamental system of equations.
It should be noted that the loadings on the middle surface can be expressed in terms of the loadings on the upper (pt,p pts, tn) and lower (pb,psb,pbn) boundary surfaces of the shell by using the static equivalence principle, as follows:
0 0 0 0 0 sin sin 1 1 1 1 2 2 2 2 sin sin 1 1 1 1 2 2 2 2 sin sin 1 1 1 1 2 2 2 t b t b s s s t b n n n h h h h p p p R R R R h h h h p p p R R R R h h h h p p p R R R 0 0 0 0 0 3 0 2 sin sin 1 1 1 1 2 2 2 2 2 2 sin sin 1 1 1 1 2 2 2 2 2 2 sin 1 1 8 2 2 t b t b s s s t R h h h h h h m p p R R R R h h h h h h m p p R R R R h h h r p p R R 3 0 3 3 0 0 sin 1 1 8 2 2 sin sin 1 1 1 1 8 2 2 8 2 2 b t b s s s h h h R R h h h h h h r p p R R R R (51) where pt , pts, t n
p are the meridional, circumferential and normal forces applied to the upper surface, and b
p, b s
p , t n
p are the meridional, circumferential and normal forces applied to the lower surface.
The boundary conditions considered in this study are the fully clamped edge boundary condition (C), the simply supported edge boundary condition (S) and the free edge boundary condition (F). They assume the following form:
Clamped edge boundary condition (C): 0 s s s u u w at or 0 1 0 s s0, (52) 0 s s s u u w at s0 or s s0 0 (53) 1
Simply supported boundary condition (S): 0 u w N M P at 0 or 0 1 0 s s0, (54) 0 s s s u w Ns Ms Ps 0 at s0 or s s0 0 (55) 1 Free edge boundary condition (F):
0 s s s N N T M M P P at or 0 1, 0 (56) s s0 0 s s s s s s s N N T M M P P at s0 or s s0, 0 (57) 1 In the above Eqs.(52-57) boundary conditions, it has been assumed s0 2R0. In order to analyze
the whole shell of revolution, and not a panel, the kinematic and physical compatibility must be added to the previous external boundary conditions. They represent the condition of continuity related to displacements and internal stress resultants. Their analytical forms are proposed as follows:
Kinematic compatibility conditions along the closing meridian (s0, 2R0):
0 0 0 0 0 0 0 0 1 ( , 0) ( , ), ( , 0) ( , ), ( , 0) ( , ), ( , 0) ( , ), ( , 0) ( , ), ( , 0) ( , ), ( , 0) ( , ) s s s s s s u u s u u s w w s s s s s (58)
Physical compatibility conditions along the closing meridian (s0, 2R0):
0 0 0 0 0 0 0 0 1 ( , 0) ( , ), ( , 0) ( , ), ( , 0) ( , ), ( , 0) ( , ), ( , 0) ( , ) , ( , 0) ( , ), ( , 0) ( , ), s s s s s s s s s s s s s s N N s N N s T T s M M s M M s P P s P P s (59)
1.3.9.1 The first fundamental equilibrium equation
By substituting Eqs.(25-42) in Eq.(44) the first fundamental equation is written as follows:
